Voltage Regulation in Smart Grids DOI: http://dx.doi.org/10.5772/intechopen.85108

connections. The charging decisions are continuous, that is, X ∈ ½ � 0; 1 where "0" stands for no charging and "1" stands for full charging. According to the grid operator, γ can be partially constrained. For instance, the PEV reactive powers can

disregarded. Stage (I) should satisfy the power flow constraints, as given by

Vð Þ <sup>i</sup>;<sup>t</sup> Vð Þ <sup>j</sup>;<sup>t</sup> Yð Þ <sup>i</sup>;<sup>j</sup> cos θð Þ <sup>i</sup>;<sup>j</sup> þ δð Þ <sup>j</sup>;<sup>t</sup> � δð Þ <sup>i</sup>;<sup>t</sup>

Vð Þ <sup>i</sup>;<sup>t</sup> Vð Þ <sup>j</sup>;<sup>t</sup> Yð Þ <sup>i</sup>;<sup>j</sup> sin θð Þ <sup>i</sup>;<sup>j</sup> þ δð Þ <sup>j</sup>;<sup>t</sup> � δð Þ <sup>i</sup>;<sup>t</sup>

where PG ið Þ ;<sup>t</sup> and QG ið Þ ;<sup>t</sup> denote the generated active and reactive powers, respectively; PL ið Þ ;<sup>t</sup> and QL ið Þ ;<sup>t</sup> are the active and reactive power demands, respectively; Vð Þ <sup>i</sup>;<sup>t</sup> and δð Þ <sup>i</sup>;<sup>t</sup> denote the magnitude and angle of the voltage, respectively; I<sup>b</sup> is the set of system busses; and Yð Þ <sup>i</sup>;<sup>j</sup> and θð Þ <sup>i</sup>;<sup>j</sup> are the magnitude and angle of the

The voltage and feeder thermal limits should also hold, and thus,

Ið Þ <sup>l</sup>;<sup>t</sup> ≤ I

PG ið Þ ;<sup>t</sup> ¼ Po ið Þ ;<sup>t</sup> QG ið Þ ;<sup>t</sup> ¼ Qo ið Þ ;<sup>t</sup>

Po ið Þ ;<sup>t</sup> <sup>≤</sup> PMPPT

(

Q2 o ið Þ;<sup>t</sup> <sup>≤</sup>S<sup>2</sup>

≤

V2 ð Þ i;t Xð Þ<sup>i</sup>

!<sup>2</sup>

CAP

CAP

where Vmin and Vmax are the maximum and minimum voltage limits, that is, 0.9 and 1.1 p.u., respectively; Ið Þ <sup>l</sup>;<sup>t</sup> denotes the per-unit current through line l ∈ ℒ; ℒ is

Typically, two back-to-back power electronic converters are used to interface PEVs and PVs, that is, DC/DC and DC/AC converters. The DC/DC converter performs MPPT with PV-based DGs or controls the PEV charging. The DC/AC converter regulates the DC link voltage and is responsible for the reactive power support [7]. The power injected to a bus should be equal to the output power of the

ð Þ<sup>l</sup> is the current carrying capacity.

, ∀i ∈I DG, t

o ið Þ ;<sup>t</sup> denotes the DG maximum power available. In both PEVs and PVs,

� <sup>P</sup><sup>2</sup>

the DC/AC converter is similar to that used with Typ. 4 wind farms. Therefore, the reactive power capability limits, defined in [17], should be used as constraints. These limits depend on the converter's power rating and DC link voltage, as follows:

o ið Þ ;<sup>t</sup> � <sup>P</sup><sup>2</sup>

Vmax c ið Þ <sup>V</sup>ð Þ <sup>i</sup>;<sup>t</sup> Xð Þ<sup>i</sup> � �<sup>2</sup>

o ið Þ ;<sup>t</sup> <sup>¼</sup> <sup>0</sup>, <sup>∀</sup>i<sup>∈</sup> <sup>I</sup>PEV, t, when the PEV voltage support is

Vmin ≤Vð Þ <sup>i</sup>;<sup>t</sup> ≤Vmax, ∀i ∈Ib, t (33)

ð Þ<sup>l</sup> , ∀l ∈ ℒ, t (34)

o ið Þ;<sup>t</sup> , <sup>∀</sup><sup>i</sup> <sup>∈</sup>IDG, t (36)

o ið Þ;<sup>t</sup> , <sup>∀</sup>i∈<sup>I</sup> DG, t (37)

o ið Þ;<sup>t</sup> , <sup>∀</sup>i∈<sup>I</sup> DG, t (38)

(35)

� � ∀i ∈Ib, t (31)

� � ∀i ∈Ib, t (32)

be set to zero, that is, QPEV

PG ið Þ ;<sup>t</sup> � PL ið Þ;<sup>t</sup> ¼ ∑

QG ið Þ ;<sup>t</sup> � QL ið Þ ;<sup>t</sup> ¼ ∑

the set of system lines; and I

DG installed at that bus:

where PMPPT

64

Qo ið Þ ;<sup>t</sup> þ

j∈I<sup>b</sup>

Research Trends and Challenges in Smart Grids

j∈I<sup>b</sup>

Y-bus admittance matrix, respectively.

where Vmax c ið Þ represents the maximum converter voltage which depends on the converter DC link voltage [17, 18], So ið Þ;<sup>t</sup> denotes the DG rated power, and Xð Þ<sup>i</sup> is the total reactance of the DG filter and interfacing transformer at bus i . If the DC/AC converter increases the set point for the DC link voltage to relax Constraint (38), the DC/DC converter will operate at a high duty cycle, which decreases its efficiency [19]. Hence, the reactive power support from the DC/AC converter is limited by the DC link voltage.

The load power at a bus should be equal to the total power consumed by regular loads and PEV:

$$P\_{L(i,t)} = P\_{NL(i,t)} + P\_{o(i,t)}^{PEV} \qquad \forall i \in \mathcal{T}\_b, t \tag{39}$$

$$Q\_{L(i,t)} = Q\_{NL(i,t)} + Q\_{o(i,t)}^{\text{PEV}} \qquad \forall i \in \mathcal{T}\_b, t \tag{40}$$

where PPEV o ið Þ ;<sup>t</sup> is the PEV active power and PNL ið Þ ;<sup>t</sup> and QNL ið Þ ;<sup>t</sup> are the active and reactive powers of normal loads, respectively. The PV power profile relies mainly on solar insolation, whereas PPEV o ið Þ ;<sup>t</sup> depends on charging decisions <sup>X</sup> chð Þ<sup>i</sup> ð Þ;<sup>t</sup> , the charging power limit in kW PCH chð Þ<sup>i</sup> ð Þ;<sup>t</sup> , and the charging efficiency <sup>η</sup>CH ch i ð Þ ð Þ , as given by.

$$P\_{o(i,t)}^{\rm PEV} = \sum\_{ch\_{(i)} \in \mathcal{CH}\_{(i)}} \frac{\mathbb{X}\_{\left(ch\_{(i)}, t\right)} P\_{CH\left(ch\_{(i)}, t\right)}}{\eta\_{CH\left(ch\_{(i)}\right)} S\_{base}}, \quad \forall i \in \mathcal{T}\_{\rm PEV}, t \tag{41}$$

where Sbase is the base power for the per-unit system in kW. The charging power limit PCH is a function of the PEV battery state of charge (SOC) and is limited by the capacity of the charger, that is, PCH ≤PCharger rated . This function is dependent on the characteristics of the battery, which can be expressed as

$$P\_{CH\left(ch\_{(i)},t\right)} = f\_{\left(ch\_{(i)},t\right)}\left(\text{SOC}^{F}\_{\left(ch\_{(i)},t\right)}\right), \qquad \forall i \in \mathcal{I}\_{PEV}, ch\_{(i)}, t \tag{42}$$

where <sup>f</sup> chð Þ<sup>i</sup> ð Þ;<sup>t</sup> is the function that represents the characteristics of the PEV battery and SOCF chð Þ<sup>i</sup> ð Þ;<sup>t</sup> is the reached SOC. The relationship between the energy delivered to a PEV battery and its SOC can be given by

$$E\_{D\left(ch\_{(i)},t\right)} = E\_{BAT\left(ch\_{(i)}\right)} \times \frac{\left(\text{SOC}^{F}\_{\left(ch\_{(i)},t\right)} - \text{SOC}^{I}\_{\left(ch\_{(i)},t\right)}\right)}{\mathbf{100}}, \quad \forall i \in \mathcal{I}\_{\text{PEV}}, ch\_{(i)}, t \tag{43}$$

where EBAT ch ð Þ ð Þ<sup>i</sup> is the battery capacity in kWh and SOCI chð Þ<sup>i</sup> ð Þ;<sup>t</sup> denotes the PEV initial SOC. The SOC of different PEVs are updated according to

$$\text{SOC}^{\text{F}}\_{\left(\text{ch}\_{(i)},\text{I}\right)} = \text{SOC}^{\text{I}}\_{\left(\text{ch}\_{(i)},\text{I}\right)} + \frac{\mathbb{X}\_{\left(\text{ch}\_{(i)},\text{I}\right)} P\_{\text{CH}\left(\text{ch}\_{(i)},\text{I}\right)}\left(\frac{\Delta T}{60}\right)}{E\_{\text{BAT}\left(\text{ch}\_{(i)}\right)}}, \quad \forall i \in \mathcal{I}\_{\text{PEV}}, \text{ch}\_{(i)}, \text{t} \tag{44}$$

where ΔT is the time step to collect the system data, run the program, and implement the decisions. Similar to DGs, the injected reactive powers from the PEVs should be limited by their converter ratings and DC link voltages, as given by

$$\left(\mathbf{Q}\_{o(i,t)}^{\mathrm{PEV}}\right)^2 \le \left(\mathbf{S}\_{o(i,t)}^{\mathrm{PEV}}\right)^2 - \left(P\_{o(i,t)}^{\mathrm{PEV}}\right)^2, \qquad \forall i \in \mathcal{T}\_{\mathrm{PEV}}, t \tag{45}$$

200 ms (100 ms for the converter settling time + 100 ms for the communication latency). Lastly, the COC refines the output solution from Stage (III) to ensure that

In this section, various case studies are presented to validate the robustness and effectiveness of the optimal coordinated voltage regulation algorithm. The 38-bus 12.66-kV distribution system is used as a test system, as shown in Figure 10. The system data can be found in [21]. The system is modified to accommodate two PEV parking lots and four PV-based DGs, with power ratings as given in Figure 10. The power demands of the two parking lots are extracted from data provided by the TPA for a weekday in 2013. Both parking lots are commercial, where P1 represents a lot in the vicinity of a train station and P2 is a lot located near downtown Toronto. The total number of PEVs during a day is displayed in Figure 11. Due to confidentiality, the addresses of the real parking lots are not mentioned. The central control unit receives the desired SOCs and sends the charging decisions to all vehicles in the parking lots. An OPAL real-time simulator (RTS) is used to model the visual test network using the SimPowerSystems blockset, which is available in Simulink/ Matlab, and an ARTEMiS plug-in [22]. The network, PEV, and DG models are

max are within the standard voltage limits. The total execution time

both Vsys

Figure 10.

67

Test network with an HiL realization.

min and Vsys

Voltage Regulation in Smart Grids

3.5 Real-time results

of the coordination algorithmΔT is 5 minutes.

DOI: http://dx.doi.org/10.5772/intechopen.85108

$$\left(\mathbf{Q}\_{o(i)}^{\mathrm{PEV}} + \frac{\mathbf{V}\_{(i,t)}^2}{X\_{(i)}}\right)^2 \le \left(\frac{\mathbf{V}\_{c(i)}^{\mathrm{max}} \mathbf{V}\_{(i,t)}}{X\_{(i)}}\right)^2 - \left(\mathbf{P}\_{o(i,t)}^{\mathrm{PEV}}\right)^2, \qquad \forall i \in \mathcal{I}\_{\mathrm{PEV}}, t \tag{46}$$

where SPEV o ið Þ ;<sup>t</sup> is the rated power of the PEV converter. In addition, the final achieved SOC, that is, SOCF chð Þ<sup>i</sup> ð Þ;<sup>t</sup> , should be limited by the SOC desired by the PEV owners SOCD chð Þ<sup>i</sup> ð Þ;<sup>t</sup> :

$$\text{SOC}^{\text{F}}\_{\left(ch\_{(i)},t\right)} \leq \text{SOC}^{\text{D}}\_{\left(ch\_{(i)},t\right)}, \quad \forall i \in \mathcal{T}\_{\text{PEV}}, ch\_{(i)}, t \tag{47}$$
